suppose $\left\{\vec{a},\vec{b}\right\}$ is a linearly independent set and that $\vec{n}$ is orthogonal to both $\vec{a}$ and $\vec{b}$ how would you prove that $\left\{\vec{a},\vec{b}, \vec{n}\right\}$ is also linearly independent?
considering the equation $c_1\vec{a}+c_2\vec{b}+c_3\vec{n}=\vec{0}$ I have managed to show that $c_3=0$ is it enough to say that $c_1=c_2=0$ from the hypothesis since $\left\{\vec{a},\vec{b}\right\}$ is linearly independent ?